DION Charlotte

We investigate the multiclass classification problem where the features are event sequences. More precisely, the data are assumed to be generated by a mixture of simple linear Hawkes processes. In this new setting, the classes are discriminated by various triggering kernels. A challenge is then to build an efficient classification procedure. We derive the optimal Bayes rule and provide a twostep estimation procedure of the Bayes classifier. In the first step, the weights of the mixture are estimated. in the second step, an empirical risk minimization procedure is performed to estimate the parameters of the Hawkes processes. We establish the consistency of the resulting procedure and derive rates of convergence. Finally, the numerical properties of the data-driven algorithm are illustrated through a simulation study where the triggering kernels are assumed to belong to the popular parametric exponential family. It highlights the accuracy and the robustness of the proposed algorithm. In particular, even if the underlying kernels are misspecified, the procedure exhibits good performance.

We consider a 1-dimensional diffusion process X with jumps. The particularity of this model relies in the jumps which are driven by a multidimensional Hawkes process denoted N. This article is dedicated to the study of a nonparametric estimator of the drift coefficient of this original process. We construct estimators based on discrete observations of the process X in a high frequency framework with a large horizon time and on the observations of the process N. The proposed nonparametric estimator is built from a least squares contrast procedure on subspace spanned by trigonometric basis vectors. We obtain adaptive results that are comparable with the one obtained in the nonparametric regression context. We finally conduct a simulation study in which we first focus on the implementation of the process and then on showing the good behavior of the estimator.

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This work focuses on the nonparametric estimation of a drift function from N discrete repeated independent observations of a diffusion process over a fixed time interval [0, T ]. We study a ridge estimator obtained by the minimization of a constrained least squares contrast. The resulting projection estimator is based on the B-spline basis. Under mild assumptions, this estimator is universally consistent with respect to an integrate norm. We establish that, up to a logarithmic factor and when the estimation is performed on a compact interval, our estimation procedure reaches the best possible rate of convergence. Furthermore, we build an adaptive estimator that achieves this rate. Finally, we illustrate our procedure through an intensive simulation study which highlights the good performance of the proposed estimator in various models.

In this paper, we consider a one-dimensional diffusion process with jumps driven by a Hawkes process. We are interested in the estimations of the volatility function and of the jump function from discrete high-frequency observations in long time horizon. We first propose to estimate the volatility coefficient. For that, we introduce in our estimation procedure a truncation function that allows to take into account the jumps of the process and we estimate the volatility function on a linear subspace of L 2 (A) where A is a compact interval of R. We obtain a bound for the empirical risk of the volatility estimator and establish an oracle inequality for the adaptive estimator to measure the performance of the procedure. Then, we propose an estimator of a sum between the volatility and the jump coefficient modified with the conditional expectation of the intensity of the jumps. The idea behind this is to recover the jump function. We also establish a bound for the empirical risk for the non-adaptive estimator of this sum and an oracle inequality for the final adaptive estimator. We conduct a simulation study to measure the accuracy of our estimators in practice and we discuss the possibility of recovering the jump function from our estimation procedure.

The recent advent of modern technology has generated a large number of datasets which can be frequently modeled as functional data. This paper focuses on the problem of multiclass classification for stochastic diffusion paths. In this context we establish a closed formula for the optimal Bayes rule. We provide new statistical procedures which are built either on the plug-in principle or on the empirical risk minimization principle. We show the consistency of these procedures under mild conditions. We apply our methodologies to the parametric case and illustrate their accuracy with a simulation study through examples.

This paper is dedicated to define two new multiple change-points detectors in the case of an unknown number of changes in the mean of a signal corrupted by additive noise. Both these methods are based on the Least-Absolute Value (LAV) criterion. Such criterion is well known for improving the robustness of the procedure, especially in the case of outliers or heavy-tailed distributions. The first method is inspired by model selection theory and leads to a data-driven estimator. The second one is an algorithm based on total variation type penalty. These strategies are numerically studied on Monte-Carlo experiments.

We consider a 1-dimensional diffusion process X with jumps. The particularity of this model relies in the jumps which are driven by a multidimensional Hawkes process denoted N. This article is dedicated to the study of a nonparametric estimator of the drift coefficient of this original process. We construct estimators based on discrete observations of the process X in a high frequency framework with a large horizon time and on the observations of the process N. The proposed nonparametric estimator is built from a least squares contrast procedure on subspace spanned by trigonometric basis vectors. We obtain adaptive results that are comparable with the one obtained in the nonparametric regression context. We finally conduct a simulation study in which we first focus on the implementation of the process and then on showing the good behavior of the estimator.

Stochastic differential equations (SDEs) are useful to model continuous stochastic processes. When (independent) repeated temporal data are available, variability between the trajectories can be modeled by introducing random effects in the drift of the SDEs. These models are useful to analyse neuronal data, crack length data, pharmacokinetics, financial data, to cite some applications among other. The R package focuses on the estimation of SDEs with linear random effects in the drift. The goal is to estimate the common density of the random effects from repeated discrete observations of the SDE. The package mixedsde proposes three estimation methods: a Bayesian parametric, a frequentist parametric and a frequentist nonparametric method. The three procedures are described as well as the main functions of the package. Illustrations are presented on simulated and real data.

In this paper, we introduce a new class of processes which are diffusions with jumps driven by a multivariate nonlinear Hawkes process. Our goal is to study their long-time behavior. In the case of exponential memory kernels for the underlying Hawkes process we establish conditions for the positive Harris recurrence of the couple (X, Y), where X denotes the diffusion process and Y the piecewise deterministic Markov process (PDMP) defining the stochastic intensity of the driving Hawkes. As a direct consequence of the Harris recurrence, we obtain the ergodic theorem for X. Furthermore, we provide sufficient conditions under which the process is exponentially β−mixing.

The recent advent of modern technology has generated a large number of datasets which can be frequently modeled as functional data. This paper focuses on the problem of multiclass classification for stochastic diffusion paths. In this context we establish a closed formula for the optimal Bayes rule. We provide new statistical procedures which are built either on the plug-in principle or on the empirical risk minimization principle. We show the consistency of these procedures under mild conditions. We apply our methodologies to the parametric case and illustrate their accuracy with a simulation study through examples.

This paper presents a general methodology for nonparametric estimation of a function s related to a nonnegative real random variable X, under a constraint of type s(0) = c. Three dierent examples are investigated: the direct observations model (X is observed), the multiplicative noise model (Y = XU is observed, with U following a uniform distribution) and the additive noise model (Y = X + V is observed where V is a nonnegative nuisance variable with known density). When a projection estimator of the target function is available, we explain how to modify it in order to obtain an estimator which satises the constraint. We extend risk bounds from the initial to the new estimator. Moreover if the previous estimator is adaptive in the sense that a model selection procedure is available to perform the squared bias/variance trade-o, we propose a new penalty also leading to an oracle type inequality for the new constrained estimator. The procedure is illustrated on simulated data, for density and survival function estimation.

This thesis includes several non-parametric probability density estimation procedures.In each case the variables of interest are not directly observed, which is a major difficulty.The first part deals with a linear mixed model where repeated observations are available.The second part is concerned with stochastic differential equation models with random effects.Several trajectories are observed in continuous time over a common time interval.The third part is placed in a multiplicative noise context.The different parts of this thesis are connected by a common inverse problem context and by a discussion of a common problem. The third part is placed in a context of multiplicative noise.The different parts of this thesis are linked by a common context of inverse problem and by a common problem: the estimation of the density of a hidden variable. In the first two parts the density of one or more random effects is estimated. In the third part, the density of the original variable is reconstructed from noisy observations. Different global estimation methods are used to build efficient estimators: kernel estimators, projection estimators or estimators built by deconvolution. The selection of parameters leads to adaptive estimators and the integrated quadratic risks are increased thanks to a Talagrand concentration inequality. A simulation study of each estimator illustrates their performance. A neural dataset is studied thanks to the procedures set up for stochastic differential equations.

We consider the model Yi = XiUi, i =1,. . , n, where the Xi, the Ui and thus the Yi are all independent and identically distributed. The Xi have density f and are the variables of interest, the Ui are multiplicative noise with uniform density on [1-a, 1+a], for some 0 < a < 1, and the two sequences are independent. However, only the Yi are observed. We study nonparametric estimation of both the density f and the corresponding survival function. In each context, a projection estimator of an auxiliary function is built, from which estimator of the function of interest is deduced. Risk bounds in term of integrated squared error are provided, showing that the dimension parameter associated with the projection step has to perform a compromise. Thus, a model selection strategy is proposed in both cases of density and survival function estimation. The resulting estimators are proven to reach the best possible risk bounds. Simulation experiments illustrate the good performances of the estimators and a real data example is described.

Two adaptive nonparametric procedures are proposed to estimate the density of the random effects in a mixed-effect Ornstein-Uhlenbeck model. First an estimator using deconvolution tools is introduced, which depends on two tuning parameters to be chosen in a data-driven way. The selection of these two parameters is achieved with a Goldenshluger and Lepski's method, extended to this particular case with a new two-dimensional penalized criterion. Then, we propose a kernel estimator of the density of the random effect, with a new bandwidth selection method. For both data driven estimators, risk bounds are provided in term of integrated $\mathbb{L}^2$-error. The estimators are evaluated on simulations and show good results. Finally, these nonparametric estimators are applied to a neuronal database of interspike intervals, and are compared with a previous parametric estimation.

This thesis includes several non-parametric probability density estimation procedures.In each case the variables of interest are not directly observed, which is a major difficulty.The first part deals with a linear mixed model where repeated observations are available.The second part is concerned with stochastic differential equation models with random effects.Several trajectories are observed in continuous time over a common time interval.The third part is placed in a multiplicative noise context.The different parts of this thesis are connected by a common inverse problem context and by a discussion of a common problem. The third part is placed in a context of multiplicative noise.The different parts of this thesis are linked by a common context of inverse problem and by a common problem: the estimation of the density of a hidden variable. In the first two parts the density of one or more random effects is estimated. In the third part, the density of the original variable is reconstructed from noisy observations. Different global estimation methods are used to build efficient estimators: kernel estimators, projection estimators or estimators built by deconvolution. The selection of parameters leads to adaptive estimators and the integrated quadratic risks are increased thanks to a Talagrand concentration inequality. A simulation study of each estimator illustrates their performance. A neural dataset is studied thanks to the procedures set up for stochastic differential equations.

Relapsing polychondritis (RP) is a rare condition characterized by recurrent inflammation of cartilaginous tissue and systemic manifestations. Data on this disease remain scarce. This study was undertaken to describe patient characteristics and disease evolution, identify prognostic factors, and define different clinical phenotypes of RP. METHODS: We performed a retrospective study of 142 patients with RP who were seen between 2000 and 2012 in a single center. RESULTS: Of the 142 patients, 86 (61%) were women. The mean ± SD age at first symptoms was 43.5 ± 15 years. Patients had the following chondritis types: auricular (89%. n = 127), nasal (63%. n = 89), laryngeal (43%. n = 61), tracheobronchial (22%. n = 32), and/orcostochondritis (40%. n = 57). The main other manifestations were articular (69%. n = 98), ophthalmologic (56%. n = 80), audiovestibular (34%. n = 48), cardiac (27%. n = 38), and cutaneous (28%. n = 40). At a mean ± SD followup of 13 ± 9 years, the 5- and 10-year survival rates were 95 ± 2% and 91 ± 3%, respectively. Factors associated with death on multivariable analysis were male sex (P = 0.01), cardiac abnormalities (P = 0.03), and concomitant myelodysplastic syndrome (MDS) (P = 0.004) or another hematologic malignancy (P = 0.01). Cluster analysis revealed that separating patients into 3 groups was clinically relevant, thereby separating patients with associated MDS, those with tracheobronchial involvement, and those without the 2 features in terms of clinical characteristics, therapeutic management, and prognosis. CONCLUSION: This large series of patients with definite RP revealed an improvement in survival as compared with previous studies. Factors associated with death were male sex, cardiac involvement, and concomitant hematologic malignancy. We identified 3 distinct phenotypes.

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In this work, a mixed stochastic differential model is studied with two random effects in the drift. We assume that N trajectories are continuously observed throughout a time interval [0, T]. Two directions are investigated. First we estimate the random effects from one trajectory and give a bound of the $L^2$-risk of the estimators. Secondly, we build a nonparametric estimator of the common bivariate density of the random effects. The mean integrated squared error is studied. The performances of the density estimator are illustrated on simulations.

This paper surveys new estimators of the density of a random effect in linear mixed-effects models. Data are contaminated by random noise, and we do not observe directly the random effect of interest. The density of the noise is suposed to be known, without assumption on its regularity. However it can also be estimated. We first propose an adaptive nonparametric deconvolution estimation based on a selection method set up in Goldenshluger and Lepski (2011). Then we propose an estimator based on a simpler model selection deviced by contrast penalization. For both of them, non-asymptotic L2-risk bounds are established implying estimation rates, much better than the expected deconvolution ones. Finally the two data-driven strategies are evaluated on simulations and compared with previous proposals.